Equivalence of definitions of Frattini subgroup
This article gives a proof/explanation of the equivalence of multiple definitions for the term Frattini subgroup
View a complete list of pages giving proofs of equivalence of definitions
- 1 The definitions that we have to prove as equivalent
- 2 Facts used
- 3 Proof
- 4 References
The definitions that we have to prove as equivalent
Definition in terms of maximal subgroups
When there are no maximal subgroups, we define as the whole group
Definition as the set of nongenerators
For a group , the Frattini subgroup is defined as the set of all nongenerators. An element is termed a nongenerator if, whenever is such that generates , itself generates .
The set of nongenerators is contained in every maximal subgroup
Given: A group , a nongenerator for , a maximal subgroup of
Proof: Suppose (contradiction point) . Then, . Thus, the set is a generating set for . By the definition of nongenerator, we see that , as a set, is a generating set for . But this is a contradiction since the subgroup generated by is itself.
Thus, is forced.
Any element in the intersection of all maximal subgroups is a nongenerator
Given: A group , an element in the intersection of maximal subgroups of , a subset such that generates
To prove: generates
Proof: Let be the subgroup generated by . Suppose (contradiction point) is proper. Then, , since . Thus, is a 1-completed subgroup: it, along with one outside element, generates the whole group. By the above fact, there exists a maximal subgroup containing . But by assumption, is in every maximal subgroup, so , so , so , a contradiction.
Thus, the assumption that is proper is false, so , so generates .
- Abstract Algebra by David S. Dummit and Richard M. Foote, 10-digit ISBN 0471433349, 13-digit ISBN 978-0471433347, More info, Exercise 25, Page 199 (Section 6.2)